If we acknowledge that the more recent years are better predictors than the farther away years, then we can do better than regression. We have to make several assumptions (like an original prior, that the years are i.i.d. normal, and that weekends, holidays, and leap years don't play in the decision). Each year's prior is the previous year's posterior, and we update our estimate and our uncertainty with new data. Then the box plots of the posterior distributions for each year are on the left, and our prior (our best guess based on the data) for 2012 is on the right.
The 95% HPD credible interval for the 2012 prior is the 93.63 to 98.53 day of non leap year, or from April 3rd to April 9th. And the most likely day is April 6th. So that doesn't narrow it down too much, but thats what I get from the data.